I don't know if I can provide a definitive answer for you, but I encountered a similar problem with one of my controllers where the down directional button would constantly register as pressed and it'd make gameplay and menu selection processes an absolute nightmare.

It ended up being a dust issue and all I had to do was take it apart and go over it with a bit of a dab with a damp cotton bud and a good spray of compressed air to fix it. But I'm a bit reluctant to say that your problem is the same as mine, since I never experienced any problems with it not turning on.